Geodesic
Lipschitz minimality of the multiplication maps of unit complex, quaternion and octonion numbers
Wen, Haomin1
1Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33 Street, Philadelphia, PA 19104-6395, USA
Algebraic and Geometric Topology, Tome 14 (2014) no. 1, pp. 407-420
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We prove that the multiplication maps Sn × Sn → Sn (n = 1,3,7) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range, the unique Lipschitz constant minimizers in their homotopy classes. Other geometrically natural maps, such as projections of Hopf fibrations, have already been shown to be, up to isometries, the unique Lipschitz constant minimizers in their homotopy classes, and it is suspected that this may hold true for all Riemannian submersions of compact homogeneous spaces. Using a counterexample, we also show that being a Riemannian submersion alone without further assumptions (like homogeneity) does not guarantee the map to be the unique Lipschitz constant minimizer in its homotopy class up to isometries, even when the receiving space is just a circle.

DOI : 10.2140/agt.2014.14.407
Classification : 53C23, 53C30, 55R25, 53C43
Keywords: Lipschitz, minimizer, quaternion, octonion, Clifford algebra

Wen, Haomin  1

1 Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33 Street, Philadelphia, PA 19104-6395, USA 
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Wen, Haomin. Lipschitz minimality of the multiplication maps of unit complex, quaternion and octonion numbers. Algebraic and Geometric Topology, Tome 14 (2014) no. 1, pp. 407-420. doi: 10.2140/agt.2014.14.407

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